Abstract: In answer to a question rasied by L. Janos, it is shown (i) that is not a finite union of metrically rigid subsets; (ii) that if , then is not the union of metrically rigid subsets, where is an infinite cardinal; and (iii) that if , then is the union of metrically rigid subsets, and hence that is a countable union of metrically rigid subsets iff the continuum hypothesis holds. ( is metrically rigid iff no two distinct two-point subsets of are isometric.) Open question: assuming the continuum hypothesis, can be written as a countable union of metrically rigid subsets if ?